Respuesta :
case A) [tex]x^{2} +y^{2} -4x+12y-20=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](x^{2}-4x)+(y^{2}+12y)=20[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}-4x+4)+(y^{2}+12y+36)=20+4+36[/tex]
[tex](x^{2}-4x+4)+(y^{2}+12y+36)=60[/tex]
Rewrite as perfect squares
[tex](x-2)^{2}+(y+6)^{2}=60[/tex]
therefore
the answer case A) is
[tex]x^{2} +y^{2} -4x+12y-20=0[/tex] -----> [tex](x-2)^{2}+(y+6)^{2}=60[/tex]
case B) [tex]x^{2} +y^{2} +6x-8y-10=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](x^{2}+6x)+(y^{2}-8y)=10[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}+6x+9)+(y^{2}-8y+16)=10+9+16[/tex]
[tex](x^{2}+6x+9)+(y^{2}-8y+16)=35[/tex]
Rewrite as perfect squares
[tex](x+3)^{2}+(y-4)^{2}=35[/tex]
therefore
the answer case B) is
[tex]x^{2} +y^{2} +6x-8y-10=0[/tex] -----> [tex](x+3)^{2}+(y-4)^{2}=35[/tex]
case C) [tex]3x^{2} +3y^{2} +12x+18y-15=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](3x^{2}+12x)+(3y^{2}+18y)=15[/tex]
Factor the leading coefficient of each expression
[tex]3(x^{2}+4x)+3(y^{2}+6y)=15[/tex]
[tex](x^{2}+4x)+(y^{2}+6y)=5[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}+4x+4)+(y^{2}+6y+9)=5+4+9[/tex]
[tex](x^{2}+4x+4)+(y^{2}+6y+9)=18[/tex]
Rewrite as perfect squares
[tex](x+2)^{2}+(y+3)^{2}=18[/tex]
therefore
the answer case C) is
[tex]3x^{2} +3y^{2} +12x+18y-15=0[/tex] -----> [tex](x+2)^{2}+(y+3)^{2}=18[/tex]
case D) [tex]5x^{2} +5y^{2} -10x+20y-30=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](5x^{2}-10x)+(5y^{2}+20y)=30[/tex]
Factor the leading coefficient of each expression
[tex]5(x^{2}-2x)+5(y^{2}+4y)=30[/tex]
[tex](x^{2}-2x)+(y^{2}+4y)=6[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}-2x+1)+(y^{2}+4y+4)=6+1+4[/tex]
[tex](x^{2}-2x+1)+(y^{2}+4y+4)=11[/tex]
Rewrite as perfect squares
[tex](x-1)^{2}+(y+2)^{2}=11[/tex]
therefore
the answer case D) is
[tex]3x^{2} +3y^{2} +12x+18y-15=0[/tex] -----> [tex](x-1)^{2}+(y+2)^{2}=11[/tex]
case E) [tex]2x^{2} +2y^{2} -24x-16y-8=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](2x^{2}-24x)+(2y^{2}-16y)=8[/tex]
Factor the leading coefficient of each expression
[tex]2(x^{2}-12x)+2(y^{2}-8y)=8[/tex]
[tex](x^{2}-12x)+(y^{2}-8y)=4[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}-12x+36)+(y^{2}-8y+16)=4+36+16[/tex]
[tex](x^{2}-12x+36)+(y^{2}-8y+16)=56[/tex]
Rewrite as perfect squares
[tex](x-6)^{2}+(y-4)^{2}=56[/tex]
therefore
the answer case E) is
[tex]2x^{2} +2y^{2} -24x-16y-8=0[/tex] -----> [tex](x-6)^{2}+(y-4)^{2}=56[/tex]
case F) [tex]x^{2} +y^{2}+2x-12y-9=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](x^{2}+2x)+(y^{2}-12y)=9[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}+2x+1)+(y^{2}-12y+36)=9+1+36[/tex]
[tex](x^{2}+2x+1)+(y^{2}-12y+36)=46[/tex]
Rewrite as perfect squares
[tex](x+1)^{2}+(y-6)^{2}=46[/tex]
therefore
the answer case F) is
[tex]x^{2} +y^{2}+2x-12y-9=0[/tex] -----> [tex](x+1)^{2}+(y-6)^{2}=46[/tex]
